given a "definition" of a $\delta$-function as follows
$\int dz \, f(x,z) \, f^{-1} (y,z) = \delta (x-y)$ ,
I would like to know how to apply knowledge over this to solve an integral like
$\int dz \, \omega(x,y,z) \, f(x,z) \, f^{-1} (y,z)$ .
Integration by parts comes to mind but I am not really sure if this can be done by intpreting $f(x,z) \, f^{-1} (y,z)$ as a derivative of a delta function because of the way it works e.g. here: derivative-of-a-delta-function
Anyways I am stuck at the following point, because something obviously goes wrong
$\omega (x,y,z) \, \delta(x-y) |_{-\infty}^{\infty} -\int dz \, \partial_z \omega(x,y,z) \delta (x-y)$
Any hints on the possibilty of solving this are much appreciated.